### All Differential Equations Resources

## Example Questions

### Example Question #1 : Numerical Solutions Of Ordinary Differential Equations

Use Euler's Method to calculate the approximation of where is the solution of the initial-value problem that is as follows.

**Possible Answers:**

**Correct answer:**

Using Euler's Method for the function

first make the substitution of

therefore

where represents the step size.

Let

Substitute these values into the previous formulas and continue in this fashion until the approximation for is found.

Therefore,

### Example Question #1 : Numerical Solutions Of Ordinary Differential Equations

Approximate for with time steps and .

**Possible Answers:**

**Correct answer:**

Approximate for with time steps and .

The formula for Euler approximations .

Plugging in, we have

Here we can see that we've gotten trapped on a horizontal tangent (a failing of Euler's method when using larger time steps). As the function is not dependent on t, we will continue to move in a horizontal line for the rest of our Euler approximations. Thus .

### Example Question #1 : Euler Method

Use Euler's Method to calculate the approximation of where is the solution of the initial-value problem that is as follows.

**Possible Answers:**

**Correct answer:**

Using Euler's Method for the function

first make the substitution of

therefore

where represents the step size.

Let

Substitute these values into the previous formulas and continue in this fashion until the approximation for is found.

Therefore,

### Example Question #1 : Euler Method

Use the *implicit *Euler method to approximate for , given that , using a time step of

**Possible Answers:**

**Correct answer:**

In the implicit method, the amount to increase is given by , or in this case . Note, you can't just plug in to this form of the equation, because it's implicit: is on both sides. Thankfully, this is an easy enough form that you can solve explicitly. Otherwise, you would have to use an approximation method like newton's method to find . Solving explicitly, we have and .

Thus,

Thus, we have a final answer of

### Example Question #1 : Euler Method

Use two steps of Euler's Method with on

To three decimal places

**Possible Answers:**

4.413

4.425

4.420

4.428

4.408

**Correct answer:**

4.425

Euler's Method gives us

Taking one step

Taking another step

### Example Question #1 : Numerical Solutions Of Ordinary Differential Equations

The two-step Adams-Bashforth method of approximation uses the approximation scheme .

Given that and , use the Adams-Bashforth method to approximate for with a step size of

**Possible Answers:**

**Correct answer:**

In this problem, we're given two points, so we can start plugging in immediately. If we were not, we could approximate by using the explicit Euler method on .

Plugging into , we have

.

Note, our approximation likely won't be very good with such large a time step, but the process doesn't change regardless of the accuracy.

### Example Question #1 : Second Order Boundary Value Problems

Find the solutions to the second order boundary-value problem. , , .

**Possible Answers:**

There are no solutions to the boundary value problem.

**Correct answer:**

The characteristic equation of is , with solutions of . Thus, the general solution to the homogeneous problem is . Plugging in our conditions, we find that , so that . Plugging in our second condition, we find that and that .

Thus, the final solution is .

### Example Question #1 : Second Order Boundary Value Problems

Find the solutions to the second order boundary-value problem. , , .

**Possible Answers:**

There are no solutions to the boundary value problem.

**Correct answer:**

There are no solutions to the boundary value problem.

The characteristic equation of is with solutions of . This tells us that the solution to the homogeneous equation is . Plugging in our conditions, we find that so that . Plugging in our second condition, we have which is obviously false.

This problem demonstrates the important distinction between initial value problems and boundary value problems: Boundary value problems don't always have solutions. This is one such case, as we can't find that satisfy our conditions.

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